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b317718 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | import OAI.Probability.InvariantIsing.Core.Variational
namespace OAI
/-! Sequential L1 compactness of bounded monotone overlap paths. The
proof extracts the values on the rationals and discards only the
countable discontinuity set of the resulting monotone path. -/
noncomputable section
open MeasureTheory Set Filter
open scoped Topology
namespace InvariantIsing
private def rationalLowerValues (f : ℚ → Icc (0 : ℝ) 1) (x : ℝ) : Set ℝ :=
(fun q : ℚ => (f q : ℝ)) '' {q : ℚ | (q : ℝ) < x}
private lemma rationalLowerValues_nonempty (f : ℚ → Icc (0 : ℝ) 1) (x : ℝ) :
(rationalLowerValues f x).Nonempty := by
obtain ⟨q, hq⟩ := exists_rat_lt x
exact ⟨f q, q, hq, rfl⟩
private lemma rationalLowerValues_bdd (f : ℚ → Icc (0 : ℝ) 1) (x : ℝ) :
BddAbove (rationalLowerValues f x) := by
refine ⟨1, ?_⟩
rintro _ ⟨q, _, rfl⟩
exact (f q).property.2
private def rationalLowerPath (f : ℚ → Icc (0 : ℝ) 1) : OverlapPath where
val x := sSup (rationalLowerValues f x)
monotone := by
intro x y hxy
apply csSup_le_csSup (rationalLowerValues_bdd f y) (rationalLowerValues_nonempty f x)
rintro _ ⟨q, hq, rfl⟩
exact ⟨q, hq.trans_le hxy, rfl⟩
nonneg x := by
obtain ⟨q, hq⟩ := exists_rat_lt x
exact (f q).property.1.trans (le_csSup (rationalLowerValues_bdd f x) ⟨q, hq, rfl⟩)
le_one x := by
apply csSup_le (rationalLowerValues_nonempty f x)
rintro _ ⟨q, _, rfl⟩
exact (f q).property.2
private lemma rational_le_lowerPath (f : ℚ → Icc (0 : ℝ) 1)
{q : ℚ} {x : ℝ} (hq : (q : ℝ) < x) : (f q : ℝ) ≤ rationalLowerPath f x :=
le_csSup (rationalLowerValues_bdd f x) ⟨q, hq, rfl⟩
private lemma rationalLowerPath_tendsto (p : ℕ → OverlapPath)
(f : ℚ → Icc (0 : ℝ) 1)
(hf : ∀ q : ℚ, Tendsto (fun n => p n (q : ℝ)) atTop (𝓝 (f q : ℝ)))
(x : ℝ) (hx : ContinuousAt (rationalLowerPath f) x) :
Tendsto (fun n => p n x) atTop (𝓝 (rationalLowerPath f x)) := by
apply tendsto_order.2
constructor
· intro a ha
obtain ⟨y, hy, hay⟩ := (lt_csSup_iff (rationalLowerValues_bdd f x)
(rationalLowerValues_nonempty f x)).mp ha
obtain ⟨q, hqx, rfl⟩ := hy
filter_upwards [(hf q).eventually (Ioi_mem_nhds hay)] with n hn
exact hn.trans_le ((p n).monotone hqx.le)
· intro b hb
have hnear : {y : ℝ | rationalLowerPath f y < b} ∈ 𝓝 x :=
hx (Iio_mem_nhds hb)
obtain ⟨ε, hε, hball⟩ := Metric.mem_nhds_iff.mp hnear
let y := x + ε / 2
have hxy : x < y := by dsimp [y]; linarith
have hy : rationalLowerPath f y < b := by
apply hball
rw [Metric.mem_ball, Real.dist_eq, abs_of_pos (sub_pos.mpr hxy)]
dsimp [y]
linarith
obtain ⟨q, hxq, hqy⟩ := exists_rat_btwn hxy
have hqb : (f q : ℝ) < b := (rational_le_lowerPath f hqy).trans_lt hy
filter_upwards [(hf q).eventually (Iio_mem_nhds hqb)] with n hn
exact ((p n).monotone hxq.le).trans_lt hn
/-- Every sequence of overlap quantiles has an L1-convergent subsequence
with another admissible bounded monotone representative. -/
theorem overlapPath_ae_l1_subsequence (p : ℕ → OverlapPath) :
∃ q : OverlapPath, ∃ φ : ℕ → ℕ, StrictMono φ ∧
(∀ᵐ s ∂pathMeasure, Tendsto (fun n => p (φ n) s) atTop (𝓝 (q s))) ∧
Tendsto (fun n => ∫ s, |p (φ n) s - q s| ∂pathMeasure) atTop (𝓝 0) := by
let f : ℕ → ℚ → Icc (0 : ℝ) 1 := fun n r => ⟨p n r, (p n).nonneg r, (p n).le_one r⟩
obtain ⟨g, φ, hφ, hg⟩ := CompactSpace.tendsto_subseq f
let q := rationalLowerPath g
have hr (r : ℚ) : Tendsto (fun n => p (φ n) (r : ℝ)) atTop (𝓝 (g r : ℝ)) :=
(continuous_subtype_val.tendsto (g r)).comp ((continuous_apply r).tendsto g |>.comp hg)
have hc : ∀ᵐ s ∂pathMeasure, ContinuousAt q s := by
let : NullSingletonClass pathMeasure := by unfold pathMeasure; infer_instance
have hz := q.monotone.countable_not_continuousAt.measure_zero (μ := pathMeasure)
have hae : ∀ᵐ s ∂pathMeasure, s ∉ {x | ¬ContinuousAt q x} := compl_mem_ae_iff.mpr hz
filter_upwards [hae] with s hs
exact not_not.mp hs
have hpoint : ∀ᵐ s ∂pathMeasure, Tendsto (fun n => p (φ n) s) atTop (𝓝 (q s)) := by
filter_upwards [hc] with s hs
exact rationalLowerPath_tendsto (fun n => p (φ n)) g hr s hs
refine ⟨q, φ, hφ, hpoint, ?_⟩
have ht := tendsto_integral_of_dominated_convergence (μ := pathMeasure)
(F := fun n s => |p (φ n) s - q s|) (f := fun _ => (0 : ℝ)) (fun _ => (1 : ℝ))
(fun n => ((p (φ n)).measurable.sub q.measurable).abs.aestronglyMeasurable)
(integrable_const _) (fun n => ae_of_all _ fun s => by
rw [Real.norm_eq_abs, abs_abs]
exact abs_le.mpr ⟨by linarith [(p (φ n)).nonneg s, q.le_one s],
by linarith [(p (φ n)).le_one s, q.nonneg s]⟩) (by
filter_upwards [hpoint] with s hs
simpa only [sub_self, abs_zero] using (hs.sub_const (q s)).abs)
simpa only [integral_zero] using ht
theorem overlapPath_l1_subsequence (p : ℕ → OverlapPath) :
∃ q : OverlapPath, ∃ φ : ℕ → ℕ, StrictMono φ ∧
Tendsto (fun n => ∫ s, |p (φ n) s - q s| ∂pathMeasure) atTop (𝓝 0) := by
obtain ⟨q, φ, hφ, _, hq⟩ := overlapPath_ae_l1_subsequence p
exact ⟨q, φ, hφ, hq⟩
end InvariantIsing
end
end OAI
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